= Solution
<Fresnel diffraction> retains the quadratic transverse propagation phase; <Fraunhofer diffraction> neglects it and leaves an aperture Fourier transform. Expanding the propagation distance gives $z+y^2/(2z)+\cdots$. The aperture phase variation is of order $\pi\rho^2/(\lambda z)$, so the crossover scale is $z\sim\rho^2/\lambda$, with far-field propagation requiring a much smaller <Fresnel number> than1.
Here $\lambda\simeq0.0100\,\mathrm m$ and the full gap width gives $\rho^2/\lambda=25\,\mathrm m$, comparable with the16m range. More decisively, measured from the grazing line of sight to the far gap edge, the quadratic phase reaches $\pi(0.5)^2/(0.01\times16)=4.91\,\mathrm{rad}$. \b[Fresnel diffraction is required]; conventions using a half-width change the quoted Fresnel number but not this phase test.
Define $F(w)=C(w)+iS(w)$ with the normalization in the paper. The grazing-edge gap spans $w=0$ to
$$
w_g=0.5\sqrt{2/(\lambda z)}=1.768.
$$
The <Cornu chord power normalization> uses a received field proportional to $F(w_g)-F(0)$, while no fence gives $F(\infty)-F(-\infty)=1+i$. Numerical Fresnel integration gives $C(w_g)=0.3245$, $S(w_g)=0.4818$, and therefore
$$
\boxed{\frac{P_{\rm gap}}{P_{\rm open}}=\frac{C(w_g)^2+S(w_g)^2}{2}=0.169.}
$$
The two labelled child answers specify the corresponding vectors.
For the <largest contiguous Fresnel aperture>, a freely chosen single contiguous opening must maximize the chord between two points of the <Cornu spiral>. Its longest chord joins $F(-w_*)$ to $F(w_*)$, where the first maximum radius satisfies $C(w_*)\cos(\pi w_*^2/2)+S(w_*)\sin(\pi w_*^2/2)=0$. This gives $w_*=1.2094$ and
$$
\boxed{-0.342\,\mathrm m<x<+0.342\,\mathrm m,\qquad
P_{\max}/P_{\rm open}\simeq1.80.}
$$
Here x is signed displacement along the fence from the grazing optical axis. The <pointwise Fresnel gain versus integrated transmitted power> distinction explains why an aperture can concentrate more power at one receiver than the fully open field through constructive interference; this is not an increase of integrated transmitted power.
The <existing-opening constraint in Fresnel aperture optimization> matters literally: the original gap already occupies $[0,0.5]\,\mathrm m$. The symmetric optimum would require filling its far end. If only further removal is permitted and the final opening must be connected, maximize the chord subject to $w_-\le0$ and $w_+\ge1.768$. The constrained optimum is approximately
$$
\boxed{w_-=-1.2158,\quad w_+=2.3405,\qquad
-0.344\,\mathrm m<x<0.662\,\mathrm m,\quad P/P_{\rm open}=1.60.}
$$
These are different optimization assumptions; both are stated so the original gap is not silently replaced. The model treats the fence as an ideal opaque screen and ignores ground reflections or additional propagation paths. <NIST Digital Library of Mathematical Functions>[§7.2, normalized Fresnel integrals and their limiting values]{ref}
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1-pa2010-cornu-vectors.png]
{title=Cornu aperture vectors}
{description=Original numerical spiral with gap, open-field and two differently constrained optimum chords.}
{source=paper-1-pa2010-cornu-vectors.py}
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